Coresets for Clustering in Graphs of Bounded Treewidth

Daniel Baker, Vladimir Braverman, Lingxiao Huang, Shaofeng H.-C. Jiang, Robert Krauthgamer, Xuan Wu
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:569-579, 2020.

Abstract

We initiate the study of coresets for clustering in graph metrics, i.e., the shortest-path metric of edge-weighted graphs. Such clustering problems are essential to data analysis and used for example in road networks and data visualization. A coreset is a compact summary of the data that approximately preserves the clustering objective for every possible center set, and it offers significant efficiency improvements in terms of running time, storage, and communication, including in streaming and distributed settings. Our main result is a near-linear time construction of a coreset for k-Median in a general graph $G$, with size $O_{\epsilon, k}(\mathrm{tw}(G))$ where $\mathrm{tw}(G)$ is the treewidth of $G$, and we complement the construction with a nearly-tight size lower bound. The construction is based on the framework of Feldman and Langberg [STOC 2011], and our main technical contribution, as required by this framework, is a uniform bound of $O(\mathrm{tw}(G))$ on the shattering dimension under any point weights. We validate our coreset on real-world road networks, and our scalable algorithm constructs tiny coresets with high accuracy, which translates to a massive speedup of existing approximation algorithms such as local search for graph k-Median.

Cite this Paper


BibTeX
@InProceedings{pmlr-v119-baker20a, title = {Coresets for Clustering in Graphs of Bounded Treewidth}, author = {Baker, Daniel and Braverman, Vladimir and Huang, Lingxiao and Jiang, Shaofeng H.-C. and Krauthgamer, Robert and Wu, Xuan}, booktitle = {Proceedings of the 37th International Conference on Machine Learning}, pages = {569--579}, year = {2020}, editor = {Hal Daumé III and Aarti Singh}, volume = {119}, series = {Proceedings of Machine Learning Research}, month = {13--18 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v119/baker20a/baker20a.pdf}, url = { http://proceedings.mlr.press/v119/baker20a.html }, abstract = {We initiate the study of coresets for clustering in graph metrics, i.e., the shortest-path metric of edge-weighted graphs. Such clustering problems are essential to data analysis and used for example in road networks and data visualization. A coreset is a compact summary of the data that approximately preserves the clustering objective for every possible center set, and it offers significant efficiency improvements in terms of running time, storage, and communication, including in streaming and distributed settings. Our main result is a near-linear time construction of a coreset for k-Median in a general graph $G$, with size $O_{\epsilon, k}(\mathrm{tw}(G))$ where $\mathrm{tw}(G)$ is the treewidth of $G$, and we complement the construction with a nearly-tight size lower bound. The construction is based on the framework of Feldman and Langberg [STOC 2011], and our main technical contribution, as required by this framework, is a uniform bound of $O(\mathrm{tw}(G))$ on the shattering dimension under any point weights. We validate our coreset on real-world road networks, and our scalable algorithm constructs tiny coresets with high accuracy, which translates to a massive speedup of existing approximation algorithms such as local search for graph k-Median.} }
Endnote
%0 Conference Paper %T Coresets for Clustering in Graphs of Bounded Treewidth %A Daniel Baker %A Vladimir Braverman %A Lingxiao Huang %A Shaofeng H.-C. Jiang %A Robert Krauthgamer %A Xuan Wu %B Proceedings of the 37th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2020 %E Hal Daumé III %E Aarti Singh %F pmlr-v119-baker20a %I PMLR %P 569--579 %U http://proceedings.mlr.press/v119/baker20a.html %V 119 %X We initiate the study of coresets for clustering in graph metrics, i.e., the shortest-path metric of edge-weighted graphs. Such clustering problems are essential to data analysis and used for example in road networks and data visualization. A coreset is a compact summary of the data that approximately preserves the clustering objective for every possible center set, and it offers significant efficiency improvements in terms of running time, storage, and communication, including in streaming and distributed settings. Our main result is a near-linear time construction of a coreset for k-Median in a general graph $G$, with size $O_{\epsilon, k}(\mathrm{tw}(G))$ where $\mathrm{tw}(G)$ is the treewidth of $G$, and we complement the construction with a nearly-tight size lower bound. The construction is based on the framework of Feldman and Langberg [STOC 2011], and our main technical contribution, as required by this framework, is a uniform bound of $O(\mathrm{tw}(G))$ on the shattering dimension under any point weights. We validate our coreset on real-world road networks, and our scalable algorithm constructs tiny coresets with high accuracy, which translates to a massive speedup of existing approximation algorithms such as local search for graph k-Median.
APA
Baker, D., Braverman, V., Huang, L., Jiang, S.H., Krauthgamer, R. & Wu, X.. (2020). Coresets for Clustering in Graphs of Bounded Treewidth. Proceedings of the 37th International Conference on Machine Learning, in Proceedings of Machine Learning Research 119:569-579 Available from http://proceedings.mlr.press/v119/baker20a.html .

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